Bifurcation Analysis Of Discrete Predator-prey System With Nonlinear Harvesting

According to numbers of research articles, population model has usually been known as a complicated nonlinear system, which is an impossible task to obtain its analytical solutions, so it is necessary to solve numerical solutions or approximate solutions using numerical methods. The numerical discrete system can be obtained in the process of applying numerical method to solving population model. Due to the convergence of numerical methods, the numerical discrete system preserves the dynamical behaviors of the corresponding continuous-time model to some extent. On the other hand, population system has been dramatically affected by harvesting activities of the human being. In addition, constant-yield harvesting type and linear harvesting type might lead to some deviations. Hence, it is significantly important to study the dynamical behavior of the numerical discrete system with nonlinear harvesting on both theoretical and practical side.At the beginning, the article applied the Euler method to solve a modified Leslie-Gower predator-prey model which incorporates Holling-II and nonlinear harvesting. The dynamical behavior of the numerical discrete system was investigated. The existence and stability of the positive fixed point were studied. By using center manifold theorem and bifurcation theorem, the specific conditions for the existence of period-doubling bifurcation and Neimark-Sacker bifurcation have been derived. Secondly, the article proposed a Lotka-Volterra predator-prey system with nonlinear harvesting type. By applying the Euler scheme to the original system has turned to be a corresponding numerical discrete system. It follows the similar way as in the above problem. Stability of the positive fixed point of the numerical discrete system is studied. Some conditions for the system to undergo period-doubling bifurcation and Neimark-Sacker bifurcation are derived.After finishing the theoretical analysis in each chapter, the numerical simulation has been applied by Matlab for those corresponding situations that have been discussed. The numerical simulation indicates that the population of species has fluctuated around the critical point, which agrees with the theoreti cal analysis. Last but not least, according to the bifurcation diagram, some complicated dynamical behaviors, such as periodic orbit, attracting, and chaotic set, have been determined as extras.